A text book on Engineering Mechanics.pdf

A Text Book on

For Degree Courses of All Branches of Engineering For Universities, Engineering Colleges, Technical Universities, Technical Institutions, IITs and NITs of the Country

Useful For gate, psus & UPSC EXAMINATION

by

Dr. U.C. Jindal M.Tech, Ph.D. Former Professor & Head of the Department Department of Mechanical Engineering Delhi College of Engineering, Delhi

MADE EASY

MADE EASY Publications Corporate Office: 44-A/1, Kalu Sarai, New Delhi-110 016  |  Website: www.madeeasypublications.org Contact: 011-45124612, 9958995830, 8860378007

MADE EASY

MADE EASY Publications Corporate Office: 44-A/1, Kalu Sarai, New Delhi-110016 Website: www.madeeasypublications.org  |  Phone: 011-45124612, 09958995830 E-mail: [email protected]

Engineering Mechanics : Dr. U.C. Jindal Copyright © 2013, by MADE EASY Publications. All rights are reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photo-copying, recording or otherwise), without the prior written permission of the above mentioned publisher of this book.

First Edition: 2013 ISBN: 978-93-81069-12-7

Published by: MADE EASY Publications, New Delhi-110016

Dedicated to My Students

PREFACE At the outset I must thank Mr. B. Singh (Chairman and Managing Director of MADE EASY Group) who has very kindly given me an opportunity to serve the student community through the publication of this book on Engineering Mechanics. The present book has been designed as a self study book taking into account the severe shortage of technical teachers in engineering colleges and technical institutions. The text in the book is well explained through examples supplemented by self explanatory illustrations, exercises supplemented by hints, key points to remember, thought provoking multiple choice questions, special problems so that a student can learn this basic subject in the shortest possible time. The book covers all the syllabi in Engineering Mechanics of all the universities, IITs, NITs, deemed universities. Students appearing in competitive examinations of UPSC, GATE, PSUs will find the book as an asset to them. The book also serves the purpose of AMIE students. The book will greatly help the students who could not grasp the subject in the class room. Any suggestion for the improvement in the text of the book will be thankfully acknowledged.

March, 2013

Dr. U. C. Jindal Author

CONTENTS 1.

Vector Quantities in Mechanics......................................................... 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Introduction................................................................................................................... 1 Position Vector.............................................................................................................. 1 Displacement Vector..................................................................................................... 2 Force Vector.................................................................................................................. 3 Moment of a Force about a Point.................................................................................. 4 Varignon’s Theorem...................................................................................................... 6 Moment of a Force about an Axis................................................................................. 7 Couple and its Moment................................................................................................. 9 Component of a Couple about a Line......................................................................... 11 Problems...................................................................................................................... 13 Key points to Remember............................................................................................. 16 Practice Problems........................................................................................................ 17 Special Problems......................................................................................................... 19 Multiple Choice Questions.......................................................................................... 20

2. Equivalent Coplanar Force Systems............................................ 22 2.1 2.2 2.3 2.4 2.5

Introduction................................................................................................................. 22 Resultant of a Force by an Equivalent Couple and a Force at any Other Point.......... 24 Resultant of a Coplanar Force System........................................................................ 25 Resultant of a Parallel Forces Systems....................................................................... 26 System of Distributed Forces...................................................................................... 28 Problems...................................................................................................................... 30 Key Points to Remember............................................................................................ 32 Practice Problems........................................................................................................ 33 Special Problems......................................................................................................... 34 Multiple Choice Questions.......................................................................................... 35

3. Equilibrium of Rigid Bodies................................................................ 38 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction................................................................................................................. 38 Support Reactions....................................................................................................... 38 Free body Diagrams.................................................................................................... 40 Conditions of Equilibrium.......................................................................................... 42 System of Concurrent Coplanar Forces...................................................................... 44 Coplanar Force Systems.............................................................................................. 45 Equilibrium of a Two Force Body............................................................................... 47 Equilibrium of a Three Force Body............................................................................ 48

viii 3.9 3.10 3.11

Lami’s Theorem.......................................................................................................... 50 Two Parallel Forces..................................................................................................... 51 Several Parallel Forces in a Plane............................................................................... 53 Problems...................................................................................................................... 54 Key Points to Remember............................................................................................ 60 Practice Problems........................................................................................................ 61 Special Problems......................................................................................................... 63 Multiple Choice Questions.......................................................................................... 64

4. Plane Trusses.......................................................................................... 67 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Introduction................................................................................................................. 67 Types of Plane Trusses................................................................................................ 67 Determination of Support Reactions........................................................................... 68 Determination of Forces in Members of the Truss..................................................... 70 Method of Joints.......................................................................................................... 71 Graphical Method....................................................................................................... 72 Method of Sections..................................................................................................... 75 Wind Loads on a Truss................................................................................................ 78 Practice Problems........................................................................................................ 85 Special Problems......................................................................................................... 87 Multiple Choice Questions.......................................................................................... 88

5. Centroid and Center of Gravity...................................................... 96 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Introduction................................................................................................................. 96 Location of Centroid................................................................................................... 96 Centroid of Simple Sections....................................................................................... 99 Centroid of Area Covered under a Curve.................................................................. 103 Centroid of Composite Section................................................................................. 107 Centre of Gravity of 3 Dimensional Bodies.............................................................. 109 Centre of Gravity of Composite Bodies.................................................................... 111 Theorems of Pappus-Guldinus.................................................................................. 112 Problems.................................................................................................................... 113 Key Points to Remember.......................................................................................... 117 Practice Problems...................................................................................................... 118 Special Problems....................................................................................................... 119 Multiple Choice Questions........................................................................................ 121

6. Beams......................................................................................................... 124 6.1 6.2 6.3

Introduction............................................................................................................... 124 Shear Force in a Beam Section................................................................................. 124 Bending Moment on a Beam Section....................................................................... 126

ix 6.4 6.5 6.6

Uniformly distributed Load on a Beam.................................................................... 128 Variable Load on Beam............................................................................................. 130 Turning Moment Applied on a Beam........................................................................ 131 Problems.................................................................................................................... 132 Key Points to Remember.......................................................................................... 134 Practice Problems...................................................................................................... 134 Multiple choice questions......................................................................................... 135

7. Spatial Forces....................................................................................... 138 7.1 7.2 7.3 7.4 7.5

8.

Introduction............................................................................................................... 138 Rectangular Components of a Force in Space.......................................................... 138 A Force Passing Through Two Points....................................................................... 140 Resultant of Forces in Space..................................................................................... 141 General Case of Three Dimensional Forces.............................................................. 143 Problems.................................................................................................................... 144 Key Points to Remember.......................................................................................... 145 Practice Problems...................................................................................................... 146 Special Problems....................................................................................................... 147 Multiple choice Questions........................................................................................ 148

Friction..................................................................................................... 150

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12

Introduction............................................................................................................... 150 Laws of Dry Friction or Coulomb’s Friction............................................................ 151 Angle of Friction....................................................................................................... 152 Block on a Horizontal Plane..................................................................................... 154 Two Mating Blocks................................................................................................... 155 Cylinder Resting on a Floor and Against a Vertical Wall.......................................... 156 Angle of Repose........................................................................................................ 158 A Force Applied on a Body Lying on an Inclined Plane........................................... 158 Bodies on Inclined Plane Connected Through a Link.............................................. 160 Ladder....................................................................................................................... 162 Wedge........................................................................................................................ 164 Friction in Wheels..................................................................................................... 165 Problems.................................................................................................................... 168 Key Points to Remember.......................................................................................... 171 Practice Problems...................................................................................................... 171 Special Problems....................................................................................................... 173 Multiple Choice Questions........................................................................................ 173

9. Application of Friction in Machines............................................. 178 9.1

Introduction............................................................................................................... 178

x 9.2 9.3 9.4 9.5 9.6

Screw Threads........................................................................................................... 178 Flat Belt Drive........................................................................................................... 182 Types of Flat Belt Drives.......................................................................................... 186 V-Belt and Rope Drive.............................................................................................. 187 Rolling Resistance..................................................................................................... 188 Problems.................................................................................................................... 190 Key Points to Remember.......................................................................................... 193 Practice Problems...................................................................................................... 194 Special Problems....................................................................................................... 195 Multiple Choice Questions........................................................................................ 195

10. Second Moment of Area  (Moment of Inertia)........................... 197 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

11.

Introduction............................................................................................................... 197 Rectangular Section.................................................................................................. 197 Triangular Section..................................................................................................... 198 Circular Section......................................................................................................... 199 Polar Moment of Inertia............................................................................................ 200 Parallel Axes Theorem.............................................................................................. 202 Moment of Inertia of a Section Enclosed by a Curve............................................... 204 Principal Axes and Product of Inertia....................................................................... 206 Parallel Axes Theorem for Product of Inertia........................................................... 208 Problems.................................................................................................................... 210 Key Points to Remember.......................................................................................... 211 Practice Problems...................................................................................................... 212 Special Problems....................................................................................................... 213 Multiple Choice Questions........................................................................................ 213

Virtual Work.......................................................................................... 216

11.1 11.2 11.3 11.4 11.5 11.5

Introduction............................................................................................................... 216 Virtual Work.............................................................................................................. 216 Work Done by a Couple............................................................................................ 217 Principle of Virtual Work Applied to Beams............................................................. 218 Potential Energy and Stability................................................................................... 221 Multiple Degrees of Freedom................................................................................... 225 Problems.................................................................................................................... 227 Key Points to Remember.......................................................................................... 228 Practice Problems...................................................................................................... 229 Special Problems....................................................................................................... 230 Multiple Choice Questions........................................................................................ 231

xi 12. Rectilinear Motion.............................................................................. 235 12.1 12.2 12.3 12.4 12.5 12.6

Introduction............................................................................................................... 235 Rectilinear Motion with Constant Acceleration........................................................ 236 Motion Under Gravity............................................................................................... 237 Acceleration as a Function of Time.......................................................................... 239 Acceleration as a Function of Velocity..................................................................... 241 Acceleration as a Function of Displacement............................................................. 242 Problems.................................................................................................................... 244 Key Points to Remember.......................................................................................... 247 Practice Problems...................................................................................................... 248 Special Problems....................................................................................................... 248 Multiple Choice Questions........................................................................................ 249

13. Curvilinear Motion.............................................................................. 252 13.3 13.4 13.5 13.1 13.2

Hodograph................................................................................................................. 253 Rectangular Co-ordinates.......................................................................................... 254 Tangential and Normal Co-ordinates........................................................................ 256 Introduction............................................................................................................... 252 Plane Curvilinear Motion.......................................................................................... 252 Problems.................................................................................................................... 259 Key Points to Remember.......................................................................................... 263 Practice Problems...................................................................................................... 264 Special Problems....................................................................................................... 265 Multiple Choice Questions........................................................................................ 266

14. Projectiles............................................................................................. 269 14.1 14.2 14.2 14.3 14.4

Introduction............................................................................................................... 269 Flight of a Projectile.................................................................................................. 269 Trajectory of a Projectile........................................................................................... 271 Motion of a Body Thrown Horizontally................................................................... 273 Range of a Projectile along an Inclined Plane.......................................................... 274 Problems.................................................................................................................... 277 Key Points to Remember.......................................................................................... 280 Practice Problems...................................................................................................... 281 Special Problems....................................................................................................... 281 Multiple Choice Questions........................................................................................ 282

15. Relative Velocity................................................................................. 284 15.1 15.2

Introduction............................................................................................................... 284 Motion Relative to a Frame in Translation............................................................... 284

xii 15.3

16.

Absolute and Relative Velocity in Plane Motion...................................................... 287 Problems.................................................................................................................... 289 Key Points to Remember.......................................................................................... 294 Practice Problems...................................................................................................... 295 Special Problems....................................................................................................... 296 Multiple Choice Questions........................................................................................ 297

Motion of Rotation.............................................................................. 301

16.1 16.2 16.3 16.4

Introduction............................................................................................................... 301 Rotation about a Fixed Axis...................................................................................... 302 Various Types of Motion of Rotation........................................................................ 304 Instantaneous Centre of Rotation.............................................................................. 308 Problems.................................................................................................................... 311 Key Points to Remember.......................................................................................... 313 Practice Problems...................................................................................................... 314 Special Problems....................................................................................................... 315 Multiple Choice Questions........................................................................................ 316

17. Kinetics of a Particle......................................................................... 318 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8

Introduction............................................................................................................... 318 Newton’s Second Law of Motion............................................................................. 318 Types of Problems..................................................................................................... 319 Motion of Two Bodies in Contact............................................................................. 322 Tangential and Normal Forces in Curvilinear Motion.............................................. 327 Constrained Motion................................................................................................... 328 Motion of a Bicycle Rider Along a Circular Track.................................................. 329 Motion of a Vehicle on a Level Circular Road......................................................... 330 Problems.................................................................................................................... 332 Key points to Remember........................................................................................... 336 Practice Problems...................................................................................................... 336 Special Problems....................................................................................................... 337 Multiple Choice Questions........................................................................................ 338

18. Dynamics of Rigid Bodies in Translation.................................... 341 18.1 18.2

Introduction............................................................................................................... 341 D’ Alembert’s Principle............................................................................................. 342 Problems.................................................................................................................... 347 Key Points to Remember.......................................................................................... 350 Practice Problems...................................................................................................... 351 Special Problems....................................................................................................... 351 Multiple Choice Questions........................................................................................ 352

xiii 19.

20.

Motion of Connected Bodies........................................................... 355

19.1

Introduction............................................................................................................... 355 Key Points to Remember.......................................................................................... 361 Special Problems....................................................................................................... 361 Multiple Choice Questions........................................................................................ 362

Mass Moment of Inertia..................................................................... 364

20.1 20.2 20.3 20.4

Introduction............................................................................................................... 364 Moment of Inertia with Respect to a Co-ordinate Axis............................................ 365 Parallel-Axes Theorem.............................................................................................. 367 Moment of Inertia of Thin Plates.............................................................................. 370 Problems.................................................................................................................... 371 Key Points to Remember.......................................................................................... 374 Practice Problems...................................................................................................... 374 Multiple Choice Questions........................................................................................ 376

21. Dynamics of Rigid Bodies in Plane Motion.................................. 378 21.1

Introduction............................................................................................................... 378 Problems.................................................................................................................... 383 Key Points to Remember.......................................................................................... 388 Practice Problems...................................................................................................... 388 Multiple Choice Questions........................................................................................ 390

22. Rotation about Fixed Axis................................................................. 394 22.1 22.2 22.3 22.4

23.

Introduction............................................................................................................... 394 Rotation about Centroidal Axis................................................................................. 396 Inertia Force Method................................................................................................. 397 Centre of Percussion................................................................................................. 399 Problems.................................................................................................................... 401 Key Points to Remember.......................................................................................... 404 Practice Problems...................................................................................................... 404 Special Problems....................................................................................................... 406 Multiple Choice Questions........................................................................................ 407

Work and Energy.................................................................................. 410

23.1 23.2 23.3 23.4 23.5

Introduction............................................................................................................... 410 Work done by a Force during Sliding Motion.......................................................... 414 Work done by a Moment during Motion of Rotation............................................... 415 Change of Potential Energy in a Spring.................................................................... 417 Rolling Motion under Gravity................................................................................... 418 Problems.................................................................................................................... 419

xiv

Practice Problems...................................................................................................... 424 Special Problems....................................................................................................... 425 Multiple Choice Questions........................................................................................ 426

24. Impulse and Momentum...................................................................... 430 24.1 24.2 24.3 24.4

Introduction............................................................................................................... 430 Angular Impulse and Angular Momentum Principle................................................ 432 Pulse of Water Jet on a Vane..................................................................................... 433 Pile and Pile Hammer............................................................................................... 436 Problems.................................................................................................................... 437 Key Points to Remember.......................................................................................... 444 Practice Problems...................................................................................................... 444 Special Problems....................................................................................................... 445 Multiple Choice Questions........................................................................................ 446

25. Impact: Collision of Elastic Bodies.............................................. 448 25.1 25.2 25.3 25.4 25.5

Introduction............................................................................................................... 448 Coefficient of Restitution.......................................................................................... 449 Oblique Central Impact............................................................................................. 451 Impact of a Body on another Stationery Body.......................................................... 452 Loss of Kinetic Energy during Collision.................................................................. 453 Problems.................................................................................................................... 455 Key Points to Remember.......................................................................................... 462 Practice Problems...................................................................................................... 462 Special Problems....................................................................................................... 463 Multiple Choice Questions........................................................................................ 464

Index .......................................................................................................... 468